How to square numbers in your head
Squaring is the best first "impressive" skill in mental math: one method handles every two-digit number, and a few shortcuts make the common cases instant.
The general method: round up, round down
Move the number to the nearest multiple of ten, move it the same distance the other way, multiply those two, and add back the distance squared.
57² distance 3 = 60 × 54 + 3² = 3240 + 9 = 3249 96² distance 4 = 100 × 92 + 4² = 9200 + 16 = 9216Why it works: (n + d)(n − d) = n² − d², so n² = (n + d)(n − d) + d². You're choosing d so that one factor is a round number.
Shortcut: numbers ending in 5
Multiply the tens digit by the next number up, then write 25 after it.
35² → 3 × 4 = 12 → 1225 75² → 7 × 8 = 56 → 5625Shortcut: numbers near 50
For 50 + d, the answer starts at 25 + d hundreds and ends in d².
Shortcut: numbers near 100
For numbers just below 100, subtract the distance from the number for the first part, and write the distance squared as two digits for the last part.
97² → 97 − 3 = 94, 3² = 09 → 9409 104² → 104 + 4 = 108, 4² = 16 → 10816Squares worth memorising
| n | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 25 |
|---|---|---|---|---|---|---|---|---|---|---|
| n² | 121 | 144 | 169 | 196 | 225 | 256 | 289 | 324 | 361 | 625 |
These come up constantly, both as answers and as the d² step inside bigger problems. The full list of square numbers goes all the way to 100².
Practise it
R3CK0N's Secret mode has two timed squaring drills — one for any two-digit square and one just for numbers ending in 5 — plus a square-root estimation drill that uses the same squares in reverse.
Drill squares